An excited ozone molecule, O3*, in the atmosphere can undergo one of the following reactions,
O3* → O3 (1) fluorescence
O3* → O + O2 (2) decomposition
O3* + M → O3 + M (3)deactivation,
where M is an inert molecule, the rate constant for the fluorescence reaction is k1, the rate constant for the decomposition reaction is k2, and the rate constant for the deactivation reaction is k3. Write a simplified expression for the fraction, X, of ozone molecules undergoing deactivation in terms of the rate constants. (Use the following as necessary: k1, k2, k3, cO for [O3*], and cM for [M].)
To find the expression for the fraction of ozone molecules undergoing deactivation, let's consider the law of mass action for the deactivation reaction:
Rate of deactivation = k3 * [O3*] * [M]
Since the rate of deactivation represents the rate at which ozone molecules undergo deactivation, we can express it as the change in concentration of O3* over time:
Rate of deactivation = -d[O3*]/dt
Now, rearranging the equation:
k3 * [O3*] * [M] = -d[O3*]/dt
Since we want to find the fraction, X, of ozone molecules undergoing deactivation, we can rewrite the equation as:
(Fraction of deactivation) = (Rate of deactivation) / (Rate of deactivation + Rate of fluorescence + Rate of decomposition)
(Fraction of deactivation) = k3 * [O3*] * [M] / (k3 * [O3*] * [M] + k1 * [O3*] + k2 * [O3*])
Simplifying the expression by factoring out [O3*] from each term:
(Fraction of deactivation) = [O3*] * (k3 * [M]) / ([O3*] * (k3 * [M] + k1 + k2))
Canceling out the [O3*] term:
(Fraction of deactivation) = k3 * [M] / (k3 * [M] + k1 + k2)
Therefore, the simplified expression for the fraction, X, of ozone molecules undergoing deactivation in terms of the rate constants is:
X = k3 * [M] / (k3 * [M] + k1 + k2)